SCIENCE CHINA Information Sciences, Volume 63 , Issue 10 : 200301(2020) https://doi.org/10.1007/s11432-020-2927-2

## A new optimization algorithm applied in electromagnetics — Maxwell's equations derived optimization (MEDO)

Donglin SU 1,2,4,5,*, Lilin LI 1,3,4, Shunchuan YANG 2,4,5, Bing LI 2,4,5, Guangzhi CHEN 2,4,5, Hui XU 1,4
• AcceptedMay 25, 2020
• PublishedAug 26, 2020
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### Acknowledgment

This work was supported by National Military Key Pre-research Project of the 13th Five-Year Plan (Grant No. 41409010101), National Natural Science Foundation of China (Grant Nos. 61427803, 61771032), and Civil Aircraft Projects of China (Grant No. MJ-2017-F-11).

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• Figure 1

Coaxial model to demonstrate the interesting division of the current. The coaxial is excited by a time-varying source and ended with a matched load. The load and the source are connected by a copper with small impedance.

• Figure 2

(Color online) Schematic of MEDO. The schematic is divided into several segments. $AKG$ is the main branch of the circuit, whose impedance is marked as $Z_1$. $ACDFGK$ is the paralleled branch in current loop $1$, and $AJIHGK$ is the paralleled branch in current loop $2$. Their impedances are recorded as $Z_2$ and $Z_3$. $GH$ is the individual in the optimization algorithm whose task is to find the optimal solution. Point $G$ is the position of the individual, which is known as variable of the optimization question. $KF$ is the domain of the variable, whose minimum and maximum values are denoted as $lb$ and $ub$. $HI$ is the area to be optimized. $F_G$ and $F_A$ refer to the external force acting on $GH$, which will be introduced in detail in the body.

• Figure 3

(Color online) Schematic diagram of calculating. (a) $R_3$; (b) $L_3$; and (c) $\frac{{\rm~d}S_{CDHJ}}{{\rm~d}t}$.

• Figure 4

(Color online) Change of the accuracy of different optimizations with the increase of the dimensions when calculating. (a) $F_1$; (b) $F_2$; (c) $F_3$; (d) $F_4$; (e) $F_5$; (f) $F_6$; (g) $F_7$; (h) $F_8$.

• Figure 5

(Color online) Geometry of the 16-elements linear antenna array positioned along the $x$-axis.

• Figure 10

(Color online) Structure of the FSS being designed. (a) Array and (b) crossed dipole element.

• Figure 11

(Color online) Insert loss comparison between MEDO and GA. MEDO can obtain a wider bandwidth than GA.

• Figure 12

(Color online) Resonant frequency comparison between the TE-based, AWDO-based, and MEDO-based FDM with the analytic results. MEDO-based FDM can obtain a more accurate resonant frequency.

• Figure 15

(Color online) Extraction results of $D_1$, $D_2$, and $D_3$ obtained by different optimization algorithms in 100 times.

• Table 1

Table 1Description of 11 benchmark functions

 Function Range Dimension $F_{\rm~min}$ $F_{1}=\sum_{i=1}^{n}~(x_{i}-2)^{2}$ [$-$10,~10] 40 0 $F_{2}=\sum_{i=1}^{n}\left|x_{i}\right|+\prod_{i=1}^{n}\left|x_{i}\right|$ [$-$10,~10] 40 0 $F_{3}=\sum_{i=1}^{n}\left(\sum_{j=1}^{i}\left(x_{j}\right)\right)^{2}$ [$-$100,~100] 40 0 $F_{4}=\sum_{i=1}^{n}\left|x_{i}+0.5\right|^{2}$ [$-$100,~100] 40 0 $F_{5}={\rm~r~a~n~d~o~m}[0,1)+\sum_{i=1}^{n}~i~x_{i}^{4}$ [$-$1.28,~1.28] 40 0 $F_{6}=\sum_{i=1}^{n}\left[x_{i}^{2}-10~\cos~\left(2~\pi~x_{i}\right)+10\right]$ [$-$5.12,~5.12] 40 0 $F_{7}=-20~\exp~(-0.2~\sqrt{\frac{1}{n}~\sum_{i=1}^{n}~x_{i}^{2}})-\exp~\left(\frac{1}{n}~\sum_{i=1}^{n}~\cos~\left(2~\pi~x_{i}\right)\right)+20+e$ [$-$32,~32] 40 0 $F_{8}=\frac{1}{4000}~\sum_{i=1}^{n}~x_{i}^{2}-\prod_{i=1}^{n}~\cos~\left(\frac{x_{i}}{\sqrt{i}}\right)+1$ [$-$600,~600] 40 0 $F_{9}=[1+\left(x_{1}+x_{2}+1\right)^{2}\left(19-14~x_{1}+3~x_{1}^{2}-14~x_{2}+6~x_{1}~x_{2}+3~x_{2}^{2}\right)]~$ [$-$2,~2] 2 3 $\times~[30+\left(2~x_{1}-3~x_{2}\right)^{2}~\times\left(18-32~x_{1}+12~x_{1}^{2}+48~x_{2}-36~x_{1}~x_{2}+27~x_{2}^{2}\right)]$ $F_{10}=0.5+\frac{\sin~^{2}(x_{1}^{2}-x_{2}^{2})-0.5}{(1+0.001(x_{1}^{2}+x_{2}^{2}))^{2}}$ [$-$100,~100] 2 0 $F_{11}=-0.5+\frac{\sin~^{2}~\sqrt{x_{1}^{2}+x_{2}^{2}}}{(1+0.001(x_{1}^{2}+x_{2}^{2}))^{2}}$ [$-$100,~100] 2 $-$1
•

Algorithm 1 Time-varying effect optimization

Require:${\rm~min}~f_{\rm~objective}(x)$;

Preset: objective function, population size, maximum iterations;

Preset: $\vert~\boldsymbol~g~\vert$, $B_0$, $\rho~S_{\rm~bar}$, $R_1$, $L_2$;

Initialization: $x_0$, $v_0$;

while ${\rm~iter}~<~\text~{maximum~number~of~iterations}$ do

Calculate $f_{\rm~objective}(x)$;

$u=-\nabla~f(x)$;

Calculate $Z_{\text~{\rm~total}}$, $Z_2$, $Z_3$, $\frac{{\rm~d}~S}{{\rm~d}~t}$;

$i_{3}=\frac{\frac{u}{Z_{\text~{\rm~total~}}}~Z_{2}+B_0~\frac{{\rm~d}~S}{{\rm~d}~t}}{Z_{2}+Z_{3}}$;

$v_{\rm~new}=-g~\cdot~\text~{\rm~iter}~\cdot~v_{\rm~cur}+(\frac{B_0}{\rho~S}~i_{3}-x_{\text~{\rm~last}})~\cdot~\text~{iter}$;

$x_{\rm~new}~=~x_{\rm~cur}~+~v_{\rm~new}$;

if $\vert~x_{\rm~new}-x_{\rm~cur}\vert~<{\rm~threshold}$ then

Exit.

end if

end while

• Table 2

Table 2Values of the coefficients of MEDO when calculating different benchmark functions

 Function $B_0$ $\rho~S$ $\vert~\boldsymbol~g~\vert$ $R_1$ $L_2$ $F_{1}$ 2 5E + 2 2E $-$ 3 1E + 6 3E + 9 $F_{2}$ 0 5E + 2 2E $-$ 3 1E + 2 3E + 9 $F_{3}$ 0 5E + 2 2E $-$ 3 1E + 2 3E + 9 $F_{4}$ 5E $-$ 1 5E + 2 2E $-$ 3 3E + 2 3E + 9 $F_{5}$ 0 1E + 3 8E $-$ 4 3E + 6 8E + 9 $F_{6}$ 1.16E $-$ 9 5.63E + 2 2E $-$ 3 1E + 3 3E + 9 $F_{7}$ 0 5E + 2 2E $-$ 3 5E + 2 3E + 9 $F_{8}$ 0 5E + 2 2E $-$ 3 5E + 2 3E + 9 $F_{9}$ 1.4 9E + 3 3E $-$ 5 5E + 2 3E + 9 $F_{10}$ 2.8E $-$ 8 2.32E + 2 5E $-$ 3 5.1E + 2 3E + 5 $F_{11}$ 1E $-$ 1 2E + 2 5E $-$ 3 5E + 2 3E + 5
• Table 3

Table 3Comparison of the average of optimization results between MEDO, AWDO, PSO, DE, and GD

 Function MEDO AWDO PSO DE GD $F_1$ 2.06E $-$ 10 8.10E $-$ 09 1.159299 0.356657 0 $F_2$ 0 0 14156.03 4.753654 1E + 38 $F_3$ 0 0 5.95349 36.35761 8.16E $-$ 9 $F_4$ 2.39E $-$ 16 5.88E $-$ 10 6.799546 36.06591 1.94E $-$ 27 $F_5$ 6.08E $-$ 05 6.80E $-$ 05 65.75983 0.281079 22.54327 $F_6$ 0 0 181.557 320.1764 350.34270 $F_7$ 8.88E $-$ 16 8.88E $-$ 16 20.25262 2.878327 19.50377 $F_8$ 0 0 753.0833 1.349412 1.05304 $F_9$ 3.0001 3.0001 3.008 3 144.95598 $F_{10}$ 0 0 1.78E $-$ 08 0 0.50283 $F_{11}$ $-$0.9999997 $-$0.99922 $-$0.99998 $-$0.9966 $-$0.53942
• Table 4

Table 4Comparison of the variance of optimization results between MEDO, AWDO, PSO, DE, and GD

 Function MEDO AWDO PSO DE GD $F_1$ 2.56E $-$ 19 1.73E $-$ 15 0.094204 0.051561 0 $F_2$ 0 0 1.99E + 11 2.700029 1E + 78 $F_3$ 0 0 54.079 421.2083 1.50E $-$ 18 $F_4$ 1.16E $-$ 32 2.74E $-$ 18 42.17984 361.9966 3.61E $-$ 55 $F_5$ 4.20E $-$ 09 4.84E $-$ 09 1891.318 0.008171 2079.07924 $F_6$ 0 0 1477.551 356.0968 2886.41731 $F_7$ 0 0 0.117238 0.172126 0.01412 $F_8$ 0 0 1954.504 0.039167 5.29E $-$ 5 $F_9$ 1.81E $-$ 8 1.34E $-$ 06 0.000105 8.05E $-$ 31 80171.13443 $F_{10}$ 0 0 3.42E $-$ 16 0 0.00179 $F_{11}$ 6.59E $-$ 14 7.02E $-$ 06 3.53E $-$ 10 2.17E $-$ 05 0.00708
• Table 5

Table 5Comparison of the SLL obtained by MEDO and PSO

 Optimization SLL (dB) MEDO $-$35.71 PSO $-$31.29
• Table 6

Table 6Comparison of the SLL obtained by MEDO and RGA

 Optimization SLL (dB) Null depth ($75\degree$ and $105~\degree$) (dB) Null depth ($68\degree$ and $112~\degree$) (dB) MEDO $-$420.83 $-$61.65 $-$61.66 RGA $-$15.18 $-$56.00 $-$86.07
• Table 7

Table 7Excitation current amplitudes and element positions of 16-element linear array obtained by MEDO and PSO for Subsection sect. 4.1.1

 shortstackElement number shortstackMEDO (for SLL suppression only) shortstackMEDO (for SLL suppression and null control) shortstackPSO (for SLL suppression only) Excitation Position Excitation Position Excitation Position amplitude spacing amplitude spacing amplitude spacing $1$st 0.151317 0.62926 0.257102 0.400000 0.210000 0.55053 $2$nd 0.291622 0.80504 0.310612 0.724125 0.401480 0.66321 $3$rd 0.468041 0.81434 0.333657 0.650454 0.320900 0.61984 $4$th 0.584701 0.73644 0.510201 0.559664 0.487930 0.40000 $5$th 0.564219 0.63420 0.382737 0.529326 0.865320 0.65443 $6$th 0.69427 0.58506 0.551289 0.550030 1.000000 0.72970 $7$th 0.719992 0.60010 0.661875 0.574347 0.975160 0.73800 $8$th 0.595556 0.50668 0.685965 0.703852 0.819060 0.66374 $9$th 0.777297 0.53528 0.713194 0.708551 0.647310 0.59657 $10$th 0.762435 0.63004 0.713019 0.689588 0.637890 0.61989 $11$th 0.710614 0.63960 0.527354 0.666631 0.503760 0.72864 $12$th 0.629917 0.67286 0.618191 0.655562 0.270990 0.69000 $13$th 0.435429 0.68788 0.321751 0.532389 0.178350 0.60816 $14$th 0.330503 0.65466 0.267579 0.444508 0.067443 0.64391 $15$th 0.213359 0.68206 0.450657 0.715799 0.049728 0.73366 $16$th 0.136245 0.67226 0.423021 0.809753 0.024616 0.82829
• Table 8

Table 8Excitation current amplitude distribution for original and corrected pattern of MEDO and QPSO

 Element number Original pattern Corrected pattern by MEDO Corrected pattern by QPSO $1$st 0.2769 0.01 0.1741 $2$nd 0.3087 0.035661 0.1523 $3$rd 0.4087 0 0 $4$th 0.2993 0.151105 0.3372 $5$th 0.5075 0 0 $6$th 0.5807 0.2104 0.2692 $7$th 0.775 0.25398 0.5821 $8$th 0.7555 0.257913 0.5638 $9$th 0.7742 0.5 0.6564 $10$th 0.7219 0.520421 0.692 $11$th 0.8832 0.60911 0.8583 $12$th 0.7731 0.686959 0.7872 $13$th 0.6126 0.765111 0.864 $14$th 0.809 0.748587 0.8718 $15$th 0.5706 0.788953 0.8739 $16$th 0.5349 0.669952 0.6647 $17$th 0.5196 0.551261 0.5375 $18$th 0.3102 0.556071 0.4607 $19$th 0.2837 0.50677 0.3564 $20$th 0.1598 0.246381 0.1574
• Table 9

Table 9Parameters of the two-layers FSS structure

 Class Parameter Value Substrate $\epsilon~_r$ 4.4 Loss rangent 0.02 Structure 1 $L_1$ 9.87 mm $w_1$ 1.35 mm $h_1$ 1.57 mm $T_{x1}$ 13 mm $T_{y1}$ 13 mm Structure 2 $L_2$ 9.1 mm $w_2$ 1.77 mm $h_2$ 1.57 mm $T_{x2}$ 13 mm $T_{y2}$ 13 mm
• Table 10

Table 10Optimization results comparison between MEDO and GA

 Optimal Resonant Resonant Bandwidth gap (mm) frequency (1) (GHz) frequency (2) (GHz) ($-$10 dB) (GHz) MEDO $8.2~$ 9.75 10.53 3.09 GA $7.43$ 9.73 10.49 3.04
• Table 11

Table 11Comparison of the parameters extraction results obtained by MEDO and AWDO

 Item Preset value $(%)$ AWDO MEDO Best Variance Best Variance $D_1$ 4.00 3.94 1.03E $-$ 2 3.94 3.98E $-$ 3 $D_2$ 36.0 36.1 1.09E $-$ 2 36.1 3.96E $-$ 3 $D_3$ 5.00 4.95 1.09E $-$ 2 4.96 5.20E $-$ 3

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